Coordinate Geometry

The points Q moves such that the length of the tangent from Q to the circle
[tex]x^2+y^2+4x+8y+9=0[/tex] is equal to the distance of Q from the origin ). Determine the locus of Q.

I am basically clueless about this question...but I will try to provide as many work I have done on this question.

I assume that we are required to find the locus of point Q. I reckon that locus of Q is a curve, since the word tangent is only suitable for curves. Hence, I will have to find the gradient of the tangent of the curve in order to solve for the tangent equation, then finding the perpendicular distance of the tangent to the circle, and equate it to the distance from Q to the origin.

But I found that the above methods seems overcomplicated and not likely to be the solution. I check the answer, but to my surprise, the locus of Q is a linear equation [tex]4x+8y+9=0[/tex]. Is the answer wrong? And how should I approach this question?

Best way is to start with a picture. Work out the centre and radius of the circle, sketch it on a set of axes. Plot an arbitrary point Q. Draw the tangent from Q to the circle. Join Q to the origin. Work out the distances required and equate them.